A game with divisors and absolute differences of exponents

TL;DR

This paper introduces a novel number game involving divisors and exponents, drawing parallels to Gilbreath's conjecture and Ducci games, and provides theoretical results and open questions.

Contribution

It formulates a new number game at the exponent level, proves an analog of Gilbreath's conjecture, and explores its properties and open problems.

Findings

Proved an analog of Gilbreath's conjecture for the game

Described the evolution of the game similar to Ducci sequences

Identified open questions for further research

Abstract

In this work, we discuss a number game that develops in a manner similar to that on which Gilbreath's conjecture on iterated absolute differences between consecutive primes is formulated. In our case the action occurs at the exponent level and there, the evolution is reminiscent of that in a final Ducci game. We present features of the whole field of the game created by the successive generations, prove an analog of Gilbreath's conjecture, and raise some open questions.